三维Minkowski空间中的圆纹曲面
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摘要
在三维闵可夫斯基(Minkowski)空间中定义了以类时曲线为脊线的圆纹(canal)曲面,并对温加顿(Weingarten)圆纹曲面进行了分类.与三维欧氏空间类似,首先以类时曲线的伏雷内(Frenet)标架为基础,结合圆纹曲面的几何定义,得到了伪正交标架下以类时曲线为脊线的圆纹曲面的参数方程.然后,建立此类圆纹曲面的基本理论,包括第一、第二基本量,高斯曲率和平均曲率等.在此基础上,得到了高斯曲率和平均曲率之间的关系,并对Weingarten圆纹曲面进行了详细的讨论.得到了三维Minkowski空间中以类时曲线为脊线的Weingarten圆纹曲面是管道曲面或者旋转曲面的结论.
The canal surfaces with time-like center curves in 3D Minkowski space were defined and the Weingarten canal surfaces were classified. Similar to the studying method for surfaces in Euclidean space,at first,the parametric equation of canal surfaces under pseudo orthogonal frame was built according to the Frenet frame of time-like curves and the geometric definition of canal surfaces,then the basic theories were obtained which include two fundamental quantities,the Gaussian curvature and mean curvature and so on. Using basic theories,the relationship between the Gaussian curvature and the mean curvature were found and the Weingarten canal surfaces were studied explicitly. The conclusion was achieved that a canal surface is a Weingarten surface if and only if it is a tube or a revolution surface.
The canal surfaces with time-like center curves in 3D Minkowski space were defined and the Weingarten canal surfaces were classified. Similar to the studying method for surfaces in Euclidean space,at first,the parametric equation of canal surfaces under pseudo orthogonal frame was built according to the Frenet frame of time-like curves and the geometric definition of canal surfaces,then the basic theories were obtained which include two fundamental quantities,the Gaussian curvature and mean curvature and so on. Using basic theories,the relationship between the Gaussian curvature and the mean curvature were found and the Weingarten canal surfaces were studied explicitly. The conclusion was achieved that a canal surface is a Weingarten surface if and only if it is a tube or a revolution surface.
引文
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[10] Ro J S,Yoon D W. Tubes of Weingarten types in a Euclidean3-space[J]. Journal of the Chungcheong Mathematical Society,2009,22(3):360-366.
[2] Kim Y H,Yoon D W. On non-developable ruled surface in Lorentz-M inkow ski 3-spaces[J]. Taiwanese Journal of Mathematics,2007,11(1):197-214.
[3] Izumiya S,Takiyama A. A timelike surface in Minkowski 3-space w hich contains lightlike lines[J]. Journal of G eometry,1999,64(1):95-101.
[4] Ucum A,IlarslanK. New types of canal surfaces in M inkow ski 3-space[J]. Advances in Applied Clifford Algebras,2016,26(1):449-468.
[5] Liu H L. Ruled surfaces with lightlike ruling in 3-Minkowski space[J]. Journal of G eometry and Physics,2009,59(1):74-78.
[6] Kim Y H,Liu H L,Qian J H. Some characterizations of canal surfaces[J]. Bulletin of the Korean Mathematical Society,2016,53(2):461-477.
[7] Qian J H,Kim Y H. Classifications of canal surfaces with L1pointw ise 1-type Gauss map[J]. Milan Journal of Mathematics,2015,83(1):145-155.
[8] Doˇgan F,Yayli Y. On the curvatures of tubular surface with Bishop frame[J]. Communications,2011,60(1):59-69.
[9] Lopez R. Rotational linear Weingarten surfaces of hyperbolic type[J]. Israel Journal of Mathematics,2008,167(1):283-301.
[10] Ro J S,Yoon D W. Tubes of Weingarten types in a Euclidean3-space[J]. Journal of the Chungcheong Mathematical Society,2009,22(3):360-366.